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Mathematics

Circle

Understand a circle from its center outward. Explore how radius sets its size, why circumference and area follow different rules, and how angles describe arcs and sectors.
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What is a circle?

A circle is the set of all points in a plane at the same distance from a fixed point called the center. That distance is its radius, rr, with r>0r>0. The circle is the boundary; the filled region inside it is a disk.

Radius and diameter

A radius joins the center to the circle. A diameter joins two points on the circle through the center, spanning two radii.

d=2rr=d2d=2r\qquad r=\frac d2
Chords, arcs, and sectors

A chord joins any two points on the circle. An arc is part of the curved boundary. A sector is the region enclosed by two radii and their connecting arc, like a slice of pizza.

Parts of a circle

Compare each part using the same circle. Blue highlights the point, line, curve, angle, or region being described; gray gives context. English and Indonesian names are shown together.

Center (titik pusat)
The fixed point inside the circle. Every point on the boundary is the same distance from this point.
Radius (jari-jari)
A line segment from the center to any point on the circle. All radii of the same circle have equal length.
rr
Diameter (diameter)
A chord that passes through the center. It is the longest chord and spans two radii.
d=2rd=2r
Chord (tali busur)
A straight line segment joining two points on the circle. A chord does not have to pass through the center.
Apothem (apotema)
The perpendicular segment from the center to a chord. It meets the chord at its midpoint; the small square marks a right angle.
a2+(c2)2=r2a^2+\left(\frac c2\right)^2=r^2

Here aa is the apothem length and cc is the chord length.

Circular arc (busur)
A portion of the curved boundary between two points. The highlighted curve is a minor arc; the remaining, longer curve is the major arc.
Circumference (keliling)
The length of the entire curved boundary. An arc is only part of this complete path around the circle.
C=2πrC=2\pi r
Sector (juring)
The region enclosed by two radii and their connecting arc. The shaded slice is a quarter of the disk.
Circular segment (tembereng)
The region between a chord and its connecting arc. The shaded cap is a minor segment, with a straight chord as its base.
Tangent (garis singgung)
A straight line that touches the circle at exactly one point. It is perpendicular to the radius at the point of contact.
Central angle (sudut pusat)
An angle whose vertex is the center and whose sides are two radii. It selects the arc and sector between those radii.
α=90∘\alpha=90^\circ

Distance around, area inside

Circumference

Circumference, CC, is the length of the boundary. Every circle has the same ratio of circumference to diameter, called π\pi.

π=Cd≈3.14159\pi=\frac Cd\approx3.14159
C=πd=2πrC=\pi d=2\pi r

Doubling the radius doubles the circumference. Measure both in length units, such as centimeters.

Area

Area, AA, measures the disk inside the circle. Imagine cutting it into many thin sectors and alternating them to approach a rectangle: its height approaches the radius and its base approaches half the circumference.

A=C2⋅r=πr2A=\frac C2\cdot r=\pi r^2

Doubling the radius quadruples the area. Use square units, such as cm2\mathrm{cm}^2.

Explore the circle

Change the radius to resize the circle, or the angle to select a different sector. Try doubling the radius and compare circumference with area. Measurements are rounded to three decimal places.

Circle explorer

The dot marks the center. The dashed line spans the diameter; blue radii and the blue arc enclose the shaded sector.

Radius: 5 cm5\ \mathrm{cm}. Central angle: 90∘90^\circ.

Radius in centimeters

5 cm5\ \mathrm{cm}
1 cm1\ \mathrm{cm}10 cm10\ \mathrm{cm}

Adjust the radius in 1 cm1\ \mathrm{cm} steps.

Central angle in degrees

90∘90^\circ
0∘0^\circ360∘360^\circ

Snaps to sudut istimewa in every quadrant: multiples of 30∘30^\circ or 45∘45^\circ, including a full turn.

Diameter
d=10 cmd=10\ \mathrm{cm}
Circumference
C≈31.416 cmC\approx31.416\ \mathrm{cm}
Disk area
A≈78.54 cm2A\approx78.54\ \mathrm{cm}^2
Arc length
s≈7.854 cms\approx7.854\ \mathrm{cm}
Sector area
Asector≈19.635 cm2A_{\mathrm{sector}}\approx19.635\ \mathrm{cm}^2

Arcs and sectors

A central angle selects a fraction of a full turn. When the angle α\alpha is in degrees, the same fraction determines the arc length ss and sector area. Here 0∘≤α≤360∘0^\circ\le\alpha\le360^\circ.

s=α360∘ 2πrs=\frac{\alpha}{360^\circ}\,2\pi r
Asector=α360∘ πr2A_{\mathrm{sector}}=\frac{\alpha}{360^\circ}\,\pi r^2

Radians measure angle as arc length divided by radius. A full turn is 2π2\pi radians. For an angle θ\theta in radians:

θ=α180∘ π\theta=\frac{\alpha}{180^\circ}\,\pi
s=rθs=r\theta
Asector=12r2θA_{\mathrm{sector}}=\frac12r^2\theta

Arc length follows the curve. The sector perimeter also includes both straight radii, so for a sector smaller than a full disk:

Psector=s+2rP_{\mathrm{sector}}=s+2r
0<θ<2π0<\theta<2\pi

A circle on the coordinate plane

For a center at (h,k)(h,k), the horizontal and vertical distances to a point (x,y)(x,y) form a right triangle with hypotenuse rr. The Pythagorean theorem gives the circle equation:

(x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2

For example, a circle centered at (2,−1)(2,-1) with radius 33 has equation:

(x−2)2+(y+1)2=9(x-2)^2+(y+1)^2=9

A tangent touches the circle at a single point and is perpendicular to the radius drawn to that point. The unit circle has center at the origin and radius 11; explore its connection to rotation in Euler’s identity.

Try it yourself

Keep answers exact using π\pi until the final step. Solve each problem before revealing the solution.

From diameter to area
d=12 cmd=12\ \mathrm{cm}

Findrr,CC,AA.

A slice of a circle
r=6 cmr=6\ \mathrm{cm}
α=60∘\alpha=60^\circ

Findss,AsectorA_{\mathrm{sector}}.

Recover the radius
C=10π mC=10\pi\ \mathrm{m}

Findrr,AA.